Validating Back-links of FOLID Cyclic Pre-proofs

International audienceCyclic pre-proofs can be represented as sets of finite tree derivations with back-links. In the frame of the first-order logic with inductive definitions (FOLID), the nodes of the tree derivations are labelled by sequents and the back-links connect particular terminal nodes, referred to as buds, to other nodes labelled by a same sequent. However, only some back-links can constitute sound pre-proofs. Previously , it has been shown that special ordering and derivability conditions, defined along the minimal cycles of the digraph representing a particular normal form of the cyclic pre-proof, are sufficient for validating the back-links. In that approach, a same constraint could be checked several times when processing different minimal cycles, hence one may require additional recording mechanisms to avoid redundant computation in order to downgrade the time complexity to polynomial. We present a new approach that does not need to process minimal cycles. It based on a normal form that allows to define the validation conditions by taking into account only the root-bud paths from the non-singleton strongly connected components of its digraph

E-Cyclist: Implementation of an Efficient Validation of FOL ID Cyclic Induction Reasoning (System Description)

Checking the soundness of cyclic induction reasoning for first-order logic with inductive definitions (FOLID) is decidable but the standard checking method is based on an exponential complement operation for Büchi automata. Recently, we introduced a polynomial checking method whose most expensive steps recall the comparisons done with multiset path orderings. We describe the implementation of our method in the Cyclist prover. Referred to as E-Cyclist, it successfully checked all the proofs included in the original distribution of Cyclist. Heuristics have been devised to automatically define from the analysis of the proof derivations the ordering measures that satisfy the ordering constraints. FOLID cyclic proof derivations may also be hard to certify. E-Cyclist witnesses a strong relation between the two cyclic and well-founded induction reasonings. This opens the perspective of using the known certification methods that work for well-founded induction proofs

Récurrence noethérienne pour le raisonnement de premier ordre

National audienceLa récurrence nœthérienne est un des principes les plus généraux de raisonnement formel. Dans le cadre du raisonnement de premier ordre, nous présentons une classification de ses instances pouvant être partagées en instances basées sur des termes et des formules. Nous donnons un aperçu du raisonnement par récurrence nœthérienne basée sur des termes et sur des formules, puis nous établissons des relations entre eux. Enfin, nous présentons une méthodologie pour la certification du raisonnement basé sur des formules à l’aide de l’assistant de preuve Coq